Exponents and Polynomials - XYZ Custom Plus

754Chapter 12 Exponents and PolynomialsPractice ProblemsSimplify the following:1. 5 3Here are some further examples of exponents.EXAMPLE 14 3 = 4 ⋅ 4 ⋅ 4 = 16 ⋅ 4 = 64 Exponent 3, base 42. −2 43. (−3) 44. ​ _−​ 2 23 ​ ​​Answers1. 1252. −163. 814. ​ 4 _9 ​EXAMPLE 2EXAMPLE 3EXAMPLE 4​ −​ 3 _−3 4 = −3 ⋅ 3 ⋅ 3 ⋅ 3 = −81 Exponent 4, base 3(−2) 5 = (−2)(−2)(−2)(−2)(−2) = −3224 ​ ​​= ​ −​ 3 _4 ​ ​​ −​ 3 _Exponent 5, base −24 ​ ​= ​_916 ​ Exponent 2, base −​ _ 3QUESTION: In what way are (−5) 2 and −5 2 different? ANSWER: In the first case,the base is −5. In the second case, the base is 5. The answer to the first is 25. Theanswer to the second is −25. Can you tell why? Would there be a difference in theanswers if the exponent in each case were changed to 3?We can simplify our work with exponents by developing some properties ofexponents. We want to list the things we know are true about exponents andthen use these properties to simplify expressions that contain exponents.BProduct Property for ExponentsThe first property of exponents applies to products with the same base. We canuse the definition of exponents, as indicating repeated multiplication, to simplifyexpressions like 7 4 ⋅ 7 2 .7 4 ⋅ 7 2 = (7 ⋅ 7 ⋅ 7 ⋅ 7)(7 ⋅ 7)= (7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7)= 7 6 Note: 4 + 2 = 6As you can see, multiplication with the same base resulted in addition ofexponents. We can summarize this result with the following property.Product Property for ExponentsIf a is any real number and r and s are integers, thena r ⋅ a s = a r+sIn words: To multiply two expressions with the same base, add exponentsand use the common base.4 ​NOT FOR SALE. Copyright XYZtextbooks Last re